If I have shared a Google Doc with someone so we both can edit it, is there a way I can send that purpose a URL that, when clicked, takes that person to the Google Doc and scrolls down to a particular position? e.g.:

Put the cursor on the header itself, then look at the address bar of your browser. It should change to update the fragment (hash - #) at the end like #heading=h.abc32def4 -- this should correspond to the link created in the Table of Contents. Copy that entire url and use wherever needed.

After you've got the headings in the document and formatted as such, do Insert > Table of Contents. It will add a TOC to the document with clickable links. Note that you can remove the TOC and the links are still valid. You can also insert bookmarks rather than a TOC.

It isn't a search feature though. If you look at the link that gets generated, it isn't something you'd be able to guess or manipulate.

I just tried this with a bookmark named 'Bookmark9/10'. It didn't work.

Looking at the header example, I realized that isn't what was needed.

I inserted a link to the bookmark within the same document and then clicked on this link I saw bookmark=id.iqgsajlrwjl9
When I pasted that whole string (not just 'id.xxxxx') in after the # in the url to another document, it worked.
I note this thread goes back several years. Today is 9/11/2014.